Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
In a scatterplot, we can visually inspect the strength of the correlation, as well as the direction of the relationship
The strength of a relationship indicates how closely related two variables are to one another
The direction of a relationship indicates how one variable reacts to change in the other
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
\[ r = \frac{n\sum{(xy)} - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}} \]
\[ r = \frac{n\sum{(xy)} - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}} \]
| i | X | Y | X·Y | X² | Y² |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 1 | 4 |
| 2 | 2 | 3 | 6 | 4 | 9 |
| 3 | 3 | 6 | 18 | 9 | 36 |
| 4 | 4 | 3 | 12 | 16 | 9 |
| Σ | 10 | 14 | 38 | 30 | 58 |
\[ r = \frac{4(38) - (10)(14)}{\sqrt{[4(30) - 10^2]*[4(58) - 14^2]}} = 0.4472 \]
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
While graphical methods are useful, we often want numbers to back up our conclusions
The purpose of simple linear regression analysis is to “fit” or “draw” a linear, straight line that goes through the center of all of our points, sometimes called the line-of-best-fit
\[ (\epsilon_1)^2 + (\epsilon_2)^2 + ... + (\epsilon_n)^2 = \sum_{i=1}\epsilon^2 = SSE \]
\[ \hat{y} = a + bx \]
Agenda
1 Overview and Introduction
2 Basic Review of Linear Equations
3 Scatterplots & Graphical Methods
4 Correlation Coefficient r
5 Regression Equation
6 Conclusion
Correlation and regression both give us some helpful ways of calculating information about the relationships between two continuous variables (bivariate)
One can determine both the direction and relative strength of the relationships between the variables
One can also make a determination on predicting on variable based upon another, with the caveat that there can always be the chance of error in that prediction
Module 12 Lecture - Analysis of Covariance (ANCOVA) || Analysis of Variance